Digital Signal Processing Reference
In-Depth Information
one low-frequency sub-band
j
t
f
and three high-frequency sub-bands. Three high-
frequency sub-bands images contain horizontal, vertical and diagonal detail
information of image, such as edges or noise, respectively. While the low-frequency
sub-band
j
t
f
reveals the structure information. The low-frequency sub-band image of
coarse scale has better structure information than the low-frequency sub-band image
of fine scale. Therefore, we employ the object in the low frequency sub-band image
sequences of coarse scale as the object confidence map, similar to the object
confidence map in [13], for the low sub-band image sequences of next fine scale for
decomposition. We enforce the constraint object confidence map on the formula (1).
The proposed method of multi-scale low rank and sparse decomposition for drogue
detection can be expressed by
(
)
()
( )
(2)
j
j
j
+
1
j
j
j
j
min
rank
B
+Γ
λ
D
D
,
s t F
. .
= +
B
D
j
j
BD
0
where
B
j
and
D
j
is the background and object at scale
j
,
denotes element-wise
0,
D
j
pq
+
1
≠
0
(
)
(
)
multiply. And
is the object confidence map at scale
j
. The
Γ
D
j
+
1
=
1,
D
j
+
1
=
0
pq
pq
object confidence map is used to encourage the sparse solutions
D
j
to be located on
regions same with the object confidence map at scale
j+
1. Because of the vibration in
the drogue video image, 2D parametric transformations is introduced into formula (2)
to compensate for the background motion caused by shaking cameras on the probe, as
shown in [12, 14], which can be used to model the translation, rotation and planar
deformation of the background. That is
(
)
()
( )
(3)
min
rank
B
j
+Γ
λ
j
D
j
+
1
D
j
,
s t F
. .
j
τ
j
= +
B
j
D
j
j
j
j
BD
,
,
τ
0
where
, and
denotes the
t
-th frame after the
F
j
τ
j
=
F
j
τ
j
,
,
F
j
τ
j
F
j
τ
j
1
1
T
T
t
t
transformation parameterized by vector
τ ∈
at scale
j
, where
p
is the number of
parameters of the motion model (
p
= 6 for the affine motion and
p
= 8 for the
projective motion). The problem described in formula (3) can be solved via binary
j
R
p
t
1,
D
j
pq
+
1
≠
0
(
)
(
)
matrix. Let
, in which the 1 or 0 indicates the pixel belonging to
ID
j
+
1
=
0,
D
j
+
1
=
0
pq
pq
(
)
(
)
object or not. Obviously,
. As shown in
[12], the objects should be contiguous pieces. Spatially or temporally neighboring
image pixels should be encouraged to have similar labels. The Ising model is used to
define the neighboring energy of
I
. The spatial smooth cost is given by
and
DI
j
=
j
Γ
DD
j
+
1
j
= Γ
I
j
+
1
I
j
0
0
0
0
I
−
I
pq
'
'
pq
(
)
pq p q
,
'
'
N
∈
(
)
, which
is the set of neighboring pixels. It can be represented by
()
1
,
pq p q
,
'
'
∈
N
Avec I
where
A
is an incidence matrix of neighboring pixels. And the
can be
F
j
τ
j
=+
B
j
D
j
2
(
)
(
)
c
()
c
expressed as
, where
is the complementary set of
. Since
min
F
j
τ
j
−
B
j
I
j
I
j
I
j
F
()
matrix rank and the
l
0
-norm are not convex function, then replacing
with sum
j
rank
B
